On weakly prime-additive numbers with length $4k+3$

Abstract

If a positive integer $n$ has at least two distinct prime divisors and can be written as $n=p_1^{\alpha _1}+\dots +p_t^{\alpha _t}$, where $p_13$, there exist infinitely many weakly prime-additive numbers $n$ with $m\mid n$ and $n=p_1^{\alpha _1}+\dots +p_{t}^{\alpha _{t}}$, where $p_1,\dots ,p_t$ are distinct prime divisors of $n$ and $\alpha _1,\dots ,\alpha _t$ are positive integers.

Keywords

• weakly prime-additive numbers
• Dirichlet’s theorem
• the Chinese remainder theorem